Operators of minimal norm via modified Green's functions

Abstract

SynopsisThe exterior Dirichlet and Neumann problems can be treated very satisfactorily by using a fundamental solution which is modified by adding radiating spherical wave functions. It has been shown [3], that the coefficients of these added terms can be chosen to ensure that the associated boundary integral equation formulation of the problem was uniquely solvable and, in addition, that the modified Green's function was a least squares best approximation to the exact Green's function for the problem. Here we show that the coefficients can be chosen to ensure not only unique solvability but also minimization of the norm of the modified integral operator. This leads to a constructive method of solution. The theory is illustrated when the boundary is a sphere and when it is a perturbation of a sphere.